Introduction

One-step equations involving multiplication or division are among the simplest algebraic equations.
They help learners transition from arithmetic to algebra by introducing the idea of undoing an operation to isolate a variable.

In this article, we solve equations of the form:

• $ax = b$
• $\dfrac{x}{a} = b$

where $a$ and $b$ are numbers and $a \neq 0$.

What Is a One-Step Equation?

A one-step equation is an equation that can be solved using exactly one inverse operation.

Examples:

• $4x = 20$
• $\dfrac{x}{3} = 7$
• $-5x = 15$

Key features:

• Only one operation is applied to the variable.
• Solving requires undoing that operation.

Inverse Operations

To isolate $x$, we use the inverse (opposite) of the operation applied to it.

• If the equation uses multiplication, we undo it with division.
• If the equation uses division, we undo it with multiplication.

Examples of inverse pairs:

• Multiply by $a$ ↔ Divide by $a$
• Divide by $a$ ↔ Multiply by $a$

Solving Multiplication Equations

A multiplication equation looks like: $$ax = b$$ To solve:

1. Identify the coefficient $a$.
2. Divide both sides by $a$.
3. The result is $x = \dfrac{b}{a}$.

Example:

• Solve $6x = 18$
  • Divide both sides by $6$
  • $x = 3$

Tips:

• If $a$ is negative, dividing by $a$ still works.
• If $b$ is zero, the solution is $x = 0$.

Solving Division Equations

A division equation looks like: $$\dfrac{x}{a} = b$$ To solve:

1. Multiply both sides by $a$.
2. The result is $x = ab$.

Example:

• Solve $\dfrac{x}{4} = 9$
  • Multiply both sides by $4$
  • $x = 36$

Tips:

• Division by a negative number is allowed.
• Multiplying both sides keeps the equation balanced.

Common Mistakes to Avoid

• Forgetting the inverse operation
  Example: Trying to subtract instead of divide.
• Dropping negative signs
  Keep track of signs carefully.
• Thinking division always makes numbers smaller
  Not true: dividing by a fraction makes numbers larger.
• Multiplying only one side
  Always perform the same operation on both sides.

Practice Examples (Step-by-Step)

Here are a few worked examples to reinforce the ideas.

Example 1

Solve $5x = 40$

• Divide both sides by $5$
• $x = 8$

Example 2

Solve $\dfrac{x}{7} = -3$

• Multiply both sides by $7$
• $x = -21$

Example 3

Solve $-4x = 12$

• Divide both sides by $-4$
• $x = -3$

Calculator

Exercises

1. Solve $7x = 35$

   Solution

   $7x = 35$
   Divide by $7$: $$x = 5$$
2. Solve $\dfrac{x}{6} = 9$

   Solution

   $\dfrac{x}{6} = 9$
   Multiply by $6$: $$x = 54$$
3. Solve $-3x = 12$

   Solution

   $-3x = 12$
   Divide by $-3$: $$x = -4$$
4. Solve $\dfrac{x}{-5} = 4$

   Solution

   $\dfrac{x}{-5} = 4$
   Multiply by $-5$: $$x = -20$$
5. Solve $0.5x = 10$

   Solution

   $0.5x = 10$
   Divide by $0.5$: $$x = 20$$
6. Solve $\dfrac{x}{8} = -2$

   Solution

   $\dfrac{x}{8} = -2$
   Multiply by $8$: $$x = -16$$
7. Solve $-9x = -27$

   Solution

   $-9x = -27$
   Divide by $-9$: $$x = 3$$
8. Solve $\dfrac{x}{0.25} = 3$

   Solution

   $\dfrac{x}{0.25} = 3$
   Multiply by $0.25$: $$x = 0.75$$